Can slicers change fields based upon another slicer ?

Create a field parameter with year and financial year as the fields, and use this as the slicer. You'll also need to use it in any visuals you want to be affected.

Create a measure like

Year selector is visible =
VAR ChosenYearType =
    SELECTEDVALUE ( 'Select FY Calendar'[Value] )
VAR CurrentFieldParam =
    SELECTEDVALUE ( 'Year FY Param'[Ordinal] ) // Have to use ordinal here
VAR Result =
    SWITCH (
        TRUE (),
        ChosenYearType = "Jan - Dec"
            && CurrentFieldParam = 0, // Year
        1,
        ChosenYearType = "Apr - Mar"
            && CurrentFieldParam = 0, // FY
        1
    )
RETURN
    Result

Add this measure as a filter to the field parameter slicer, set to show only when the value is 1.